# If 8.40 kJ of heat is needed to raise the temperature of a sample of metal from 15 °C to 20 °C, how many kilojoules of heat will be required to raise the temperature of the same sample of metal from 25 °C to 40 °C?

##### 1 Answer

#### Explanation:

The trick here is to realize that because the sample of metal has **the same mass** in both cases, you can say that

#q_2 = (DeltaT_2)/(DeltaT_1) * q_1#

Here

#q_1# is the amount of heat needed to raise the temperature of the sample by#DeltaT_1 = 20^@"C" - 15^@"C"# #q_2# is the amount of heat needed to raise the temperature of the sample by#DeltaT_2 = 40^@"C" - 25^@"C"#

This equation can be found by using the fact that the heat absorbed by the metal can be calculated using the equation

#color(blue)(ul(color(black)(q = m * c * DeltaT)))#

Here

#m# is themassof the sample#c# is thespecific heatof the metal

In your case, you can say that

#q_1 = m * c * DeltaT_1#

and

#q_2 = m * c * DeltaT_2#

Divide these two equations

#q_1/q_2 = (color(red)(cancel(color(black)(m * c))) * DeltaT_1)/(color(red)(cancel(color(black)(m * c))) * DeltaT_2)#

to get

#q_2 = (DeltaT_2)/(DeltaT_1) * q_1# This equation tells you that in order to increase the temperature of the metal by a factor

#(DeltaT_2)/(DeltaT_1)# when the mass of the metal isconstant, the amount of heat supplied must alsoincreaseby a factor of#(DeltaT_2)/(DeltaT_1)# .

So, plug in your values to find

#q_2 = ((40 - 25) color(red)(cancel(color(black)(""^@"C"))))/((20-15)color(red)(cancel(color(black)(""^@"C")))) * "8.40 kJ"#

#color(darkgreen)(ul(color(black)(q_2 = "25 kJ")))#

The answer is rounded to two **sig figs**.